Geometry right triangle math help please?

Question

A right triangle has a #-cm (let's say 12) perimeter, and its hypotenuse is twice as long as its shorter leg. How would you find the lengths of all three sides of this triangle.

I'm not interested in the answer. I just want to know how you solve it.

Confucius: "Give a man a fish, you'll find him for a day. Teach a man to fish, you'll feed him for life."

Thanks!

Answer 1

ah... thats the formula for a 30-60-90 triangle.

its like this. if its 30-60-90, the sides are x for one leg, 2x for hypotenous, and xsqrt(3) for the other leg.

x+2x+xsqrt3=12

4.732x=12

x=2.536

then just put into the formulas.

legs are 2.536 and 4.392

hypotenous is 5.072

all cm of course.

make it a good day

Answer 2

Let the shorter leg be x. That means the hypotenuse, which is twice as long, would be 2x.

Using Pythagoras, the other side, let that be y, (the middle length) would be:

y² = (2x)² - x²

y² = 4x² - x²

y² = 3x²

y = √(3x²)

y = x√3

Now, we have three lengths:

SHORT = x

MIDDLE = x√3

LONG = 2x

Now, as the perimeter is 12, adding these lengths and solve for x:

x + x√3 + 2x = 12

3x + x√3 = 12

x(3 + √3) = 12

x = 12 / (3 + √3)

Now you have the length of x (the smaller side). Plug this into the values for the other lengths in terms of x.

  

Answer 3

Let's not use numbers, just letters.

Let p = perimeter

Let h = hypotenuse

Let a = shorter leg

Let b = longer leg

a + b + h = p

h = 2a

a + b + 2a = p

3a + b = p

Since it's a rt. triangle, we also know:

a^2 + b^2 = h^2

a^2 + b^2 = (2a)^2

a^2 + b^2 = 4a^2

3a^2 = b^2

b = sqrt(3a^2) = a(sqrt3)

Substitute again:

3a + a(sqrt3) = p

a(3 + sqrt3) = p

a = p/(3 + sqrt3)

If you know the value of p, you can easily find the value for a. From there, you can find both b and h.